The function \(K_W(u,v)\)  is linear in \(v\)  and conjugate-linear in \(u\) . Salomon, Shalit and Shamovich showed in [1]} that \(\mathcal {H}^2_d\)  is generated by the NC functions in (REF ), and that for any \(f\)  in \(\mathcal {H}_d^2\)  and for any \(W\)  in \(\mathcal {B}_d(n)\)
\(\mathinner {\langle {f, K_W(u, z)}\rangle }_{\mathcal {H}^2_d}=\mathinner {\langle {f(W)u, v}\rangle }_{\mathbb {C}^n}.\) 
